Modeling a Symmetrically-Notched Continuum Neurosurgical Robot With Non-Constant Curvature and Superelastic Property

A neurosurgical robot with 3-dimensional (3D) distal manipulation can increase instrument workspace for reaching peripheral regions of intracranial lesions to achieve complete lesion removal and minimal brain manipulation in a keyhole procedure. In this work, we designed a Nitinol-based symmetrically notched continuum robot based on the neurosurgical clinical constraints. A mechanics model is developed for the robot using a second order differential equation that exploits the shearing force formulation based upon the Euler $-$ Bernoulli moment-curvature equation. It affords non-constant curvature analysis, and integrates the piecewise linear material model with three stages that closely approximates the nonlinear superelastic property of Nitinol. A solution was introduced to this complex mechanics model based on the fourth order Runge-Kutta method with variable transformation, and an iterative approach. Comparison with the experimental data using our proposed model shows its high modeling accuracy in terms of the bending angle and tip position for a single symmetric notch bending as well as 2D bending motion of the robot. The significantly improved accuracy over previous models justifies the added complexity of our modeling approach and could better guide the determination of the notch parameters during robot design.